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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 6 - Matrices and Determinants문제 39
7장, 문제 39

Find the quadratic function f(x) = ax² + bx + c for which ƒ( − 2) = −4, ƒ(1) = 2, and f(2) = 0.

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Start by writing the general form of the quadratic function: \(f(x) = ax^{2} + bx + c\).
Use the given points to create a system of equations by substituting each \(x\) and \(f(x)\) value into the quadratic function:
For \(f(-2) = -4\), substitute \(x = -2\) and \(f(x) = -4\) to get: \(a(-2)^{2} + b(-2) + c = -4\).
For \(f(1) = 2\), substitute \(x = 1\) and \(f(x) = 2\) to get: \(a(1)^{2} + b(1) + c = 2\).
For \(f(2) = 0\), substitute \(x = 2\) and \(f(x) = 0\) to get: \(a(2)^{2} + b(2) + c = 0\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Its graph is a parabola, and the coefficients determine its shape and position.
추천 영상:
06:36
Solving Quadratic Equations Using The Quadratic Formula

Function Evaluation

Function evaluation involves substituting a specific input value into the function to find the corresponding output. For example, f(−2) means replacing x with −2 in the quadratic expression and calculating the result.
추천 영상:
4:26
Evaluating Composed Functions

Solving Systems of Equations

To find the coefficients a, b, and c, you set up equations based on the given function values and solve the resulting system of linear equations. This process often uses substitution or elimination methods to find the unknowns.
추천 영상:
5:48
Solving Systems of Equations - Substitution
관련 실천
교과서 질문

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{x−y+z=82y−z=−72x+3y=1The inverse of [1−1102−1230] is [33−1−2−21−4−52].\(\begin{cases}\) x - y + z = 8 \\ 2y - z = -7 \\ 2x + 3y = 1 \(\end{cases}\) \\ \(\text{The inverse of }\) \(\begin{bmatrix}\) 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \(\end{bmatrix}\) \(\text{ is }\) \(\begin{bmatrix}\) 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \(\end{bmatrix}\]\text{.}\)

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교과서 질문

In Exercises 37–44, use Cramer's Rule to solve each system. {4x−5y−6z=−1x−2y−5z=−122x−y=7\(\begin{cases}\) 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y = 7 \(\end{cases}\)

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교과서 질문

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

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교과서 질문

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)

BC + CB

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교과서 질문

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

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교과서 질문

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{w−x+2y=−3x−y+z=4−w+x−y+2z=2−x+y−2z=−4The inverse of [1−12001−11−11−120−11−2] is [00−1−1141312120−10−1]\(\begin{cases}\) w - x + 2y \(\quad\]\quad\) = -3 \\ \(\quad\[\quad\) x - y + z = 4 \\ -w + x - y + 2z = 2 \\ \(\quad\]\quad\) -x + y - 2z = -4 \(\end{cases}\) \\ \(\text{The inverse of }\) \(\begin{bmatrix}\) 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \(\end{bmatrix}\) \(\text{ is }\) \(\begin{bmatrix}\) 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \(\end{bmatrix}\)

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